Showing papers on "Normal modal logic published in 2017"

Neighborhood Semantics for Modal Logic

Abstract: Day 5: Applications and Extensions The precise topics for the course will depend in part on the interests and background of the students attending the lectures. This document contains an introduction to the course including some pointers to relevant literature. The focus is on the basics of neighborhood semantics with only pointers to literature relating to the more advanced topics. The website for the course is staff.science.uva.nl/∼epacuit/nbhd esslli.html On this website you will find the lecture slides (and possibly course notes) updated daily. Two other sources will also be useful:

Theorem Provers For Every Normal Modal Logic

Abstract: We present a procedure for algorithmically embedding problems formulated in higherorder modal logic into classical higher-order logic. The procedure was implemented as a stand-alone tool and can be used as a preprocessor for turning TPTP THF-compliant theorem provers into provers for various modal logics. The choice of the concrete modal logic is thereby specified within the problem as a meta-logical statement. This specification format as well as the underlying semantics parameters are discussed, and the implementation and the operation of the tool are outlined. By combining our tool with one or more THF-compliant theorem provers we accomplish the most widely applicable modal logic theorem prover available to date, i.e. no other available prover covers more variants of propositional and quantified modal logics. Despite this generality, our approach remains competitive, at least for quantified modal logics, as our experiments demonstrate.

Four-valued modal logic: Kripke semantics and duality

Abstract: Combining multi-valued and modal logics into a single system is a long-standing concern in mathematical logic and computer science, see for example [7] and the literature cited there. Recent work in this trend [15, 17, 14] develops modal expansions of many-valued systems that are also inconsistencytolerant, along the tradition initiated by Belnap with his “useful four-valued logic” [3]. Our contribution continues on this line, and the specific problem we address is that of defining and axiomatizing the least modal logic over the four-element Belnap lattice. The problem was inspired by [5], but our solution is quite different from (and in some respects more satisfactory than) that of [5] in that we make an extensive and profitable use of algebraic and topological techniques. In fact, our algebraic and topological analyses of the logic have, in our opinion, an independent interest and contribute to the appeal of our approach. Kripke frames provide a semantics for modal logics that is both flexible with regards to intended applications and interpretations, and highly intuitive. When the non-modal part is multi-valued, though, one may wonder whether the accessibility relation between worlds should remain two-valued or be allowed to assume the same range of truth values as the logic itself. Starting from the point of view of AI applications, [7] argues forcefully that multiple values are an appropriate and useful modeling device. This is the approach taken in [5] and here, too. Our aim is to study the least modal logic over the Belnap lattice, that is, the logic determined by the class of all Kripke frames where the accessibility relation as well as semantic valuations are four-valued.

Automating Emendations of the Ontological Argument in Intensional Higher-Order Modal Logic

Abstract: A shallow semantic embedding of an intensional higher-order modal logic (IHOML) in Isabelle/HOL is presented. IHOML draws on Montague/Gallin intensional logics and has been introduced by Melvin Fitting in his textbook Types, Tableaus and Godel’s God in order to discuss his emendation of Godel’s ontological argument for the existence of God. Utilizing IHOML, the most interesting parts of Fitting’s textbook are formalized, automated and verified in the Isabelle/HOL proof assistant. A particular focus thereby is on three variants of the ontological argument which avoid the modal collapse, which is a strongly criticized side-effect in Godel’s resp. Scott’s original work.

A New Modal Framework for Epistemic Logic

Abstract: Recent years witnessed a growing interest in non-standard epistemic logics of knowing whether, knowing how, knowing what, knowing why and so on. The new epistemic modalities introduced in those logics all share, in their semantics, the general schema of $\exists x \Box \phi$, e.g., knowing how to achieve $\phi$ roughly means that there exists a way such that you know that it is a way to ensure that $\phi$. Moreover, the resulting logics are decidable. Inspired by those particular logics, in this work, we propose a very general and powerful framework based on quantifier-free predicate language extended by a new modality $\Box^x$, which packs exactly $\exists x \Box$ together. We show that the resulting language, though much more expressive, shares many good properties of the basic propositional modal logic over arbitrary models, such as finite-tree-model property and van Benthem-like characterization w.r.t.\ first-order modal logic. We axiomatize the logic over S5 frames with intuitive axioms to capture the interaction between $\Box^x$ and know-that operator in an epistemic setting.

The modal logic of set-theoretic potentialism and the potentialist maximality principles

Abstract: We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and Lowe, including the use of buttons, switches, dials and ratchets. Among the potentialist conceptions we consider are: rank potentialism (true in all larger $V_\beta$); Grothendieck-Zermelo potentialism (true in all larger $V_\kappa$ for inaccessible cardinals $\kappa$); transitive-set potentialism (true in all larger transitive sets); forcing potentialism (true in all forcing extensions); countable-transitive-model potentialism (true in all larger countable transitive models of ZFC); countable-model potentialism (true in all larger countable models of ZFC); and others. In each case, we identify lower bounds for the modal validities, which are generally either S4.2 or S4.3, and an upper bound of S5, proving in each case that these bounds are optimal. The validity of S5 in a world is a potentialist maximality principle, an interesting set-theoretic principle of its own. The results can be viewed as providing an analysis of the modal commitments of the various set-theoretic multiverse conceptions corresponding to each potentialist account.

Negative Translations and Normal Modality.

Abstract: We discuss the behaviour of variants of the standard negative translations - Kolmogorov, Godel-Gentzen, Kuroda and Glivenko - in propositional logics with a unary normal modality. More specifically, we address the question whether negative translations as a rule embed faithfully a classical modal logic into its intuitionistic counterpart. As it turns out, even the Kolmogorov translation can go wrong with rather natural modal principles. Nevertheless, we isolate sufficient syntactic criteria ensuring adequacy of well-behaved (or, in our terminology, regular) translations for logics axiomatized with formulas satisfying these criteria, which we call enveloped implications. Furthermore, a large class of computationally relevant modal logics - namely, logics of type inhabitation for applicative functors (a.k.a. idioms) - turns out to validate the modal counterpart of the Double Negation Shift, thus ensuring adequacy of even the Glivenko translation. All our positive results are proved purely syntactically, using the minimal natural deduction system of Bellin, de Paiva and Ritter extended with additional axioms/combinators and hence also allow a direct formalization in a proof assistant (in our case Coq).

“That Will Do”: Logics of Deontic Necessity and Sufficiency

Abstract: We study a logic for deontic necessity and sufficiency (often interpreted as obligation, resp. strong permission), as originally proposed in van Benthem (Bull Sect Log 8(1):36–41, 1979). Building on earlier work in modal logic, we provide a sound and complete axiomatization for it, consider some standard extensions, and study other important properties. After that, we compare this logic to the logic of “obligation as weakest permission” from Anglberger et al. (Rev Symb Log 8(4):807–827, 2015).

Second-Order Modal Logic

Abstract: This dissertation develops an inferentialist theory of meaning. It takes as a starting point that the sense of a sentence is determined by the rules governing its use. In particular, there are two features of the use of a sentence that jointly determine its sense, the conditions under which it is coherent to assert that sentence and the conditions under which it is coherent to deny that sentence. From this starting point the dissertation develops a theory of quantification as marking coherent ways a language can be expanded and modality as the means by which we can reflect on the norms governing the assertion and denial conditions of our language. If the view of quantification that is argued for is correct, then there is no tension between second-order quantification and nominalism. In particular, the ontological commitments one can incur through the use of a quantifier depend wholly on the ontological commitments one can incur through the use of atomic sentences. The dissertation concludes by applying the developed theory of meaning to the metaphysical issue of necessitism and contingentism. Two objections to a logic of contingentism are raised and addressed. The resulting logic is shown to meet all the requirement that the dissertation lays out for a theory of meaning for quantifiers and modal operators. Second-Order Modal Logic

Possibilistic Justification Logic: Reasoning About Justified Uncertain Beliefs

Abstract: Justification logic originated from the study of the logic of proofs However, in a more general setting, it may be regarded as a kind of explicit epistemic logic In such logic, the reasons a fact is believed are explicitly represented as justification terms Traditionally, the modeling of uncertain beliefs is crucially important for epistemic reasoning Graded modal logics interpreted with possibility theory semantics have been successfully applied to the representation and reasoning of uncertain beliefs; however, they cannot keep track of the reasons an agent believes a fact This article is aimed at extending the graded modal logics with explicit justifications We introduce a possibilistic justification logic, present its syntax and semantics, and investigate its metaproperties, such as soundness, completeness, and realizability

On the expressiveness of spatial constraint systems

Abstract: Epistemic, mobile and spatial behaviour are common place in today’s distributed systems. The intrinsic epistemic nature of these systems arises from the interactions of the elements taking part of them. Most people are familiar with digital systems where users share their beliefs, opinions and even intentional lies (hoaxes). Models of those systems must take into account the interactions with others as well as the distributed quality these systems present. Spatial and mobile behaviour are exhibited by applications and data moving across (possibly nested) spaces defined by, for example, friend circles, groups, and shared folders. We therefore believe that a solid understanding of the notion of space and spatial mobility as well as the flow of epistemic information is relevant in many models of today’s distributed systems.Constraint systems (cs’s) provide the basic domains and opera- tions for the semantic foundations of the family of formal declarative models from concurrency theory known as concurrent constraint programming (ccp). Spatial constraint systems (scs’s) are algebraic structures that extend cs’s for reasoning about basic spatial and epistemic behaviour such as belief and extrusion. Both spatial and epistemic assertions can be viewed as specific modalities. Other modalities can be used for assertions about time, knowledge and even the analysis of groups among other concepts used in the specification and verification of concurrent systems.In this thesis we study the expressiveness of spatial constraint systems in the broader perspective of modal and epistemic behaviour. We shall show that spatial constraint systems are sufficiently robust to capture inverse modalities and to derive new results for modal logics. We shall show that we can use scs’s to express a fundamental epistemic behaviour such as knowledge. Finally we shall give an algebraic characterization of the notion of distributed information by means of constructors over scs’s.

Bounded game-theoretic semantics for modal mu-calculus

Abstract: We introduce a new game-theoretic semantics (GTS) for the modal mu-calculus. Our so-called bounded GTS replaces parity games with alternative evaluation games where only finite paths arise; infinite paths are not needed even when the considered transition system is infinite. The novel games offer alternative approaches to various constructions in the framework of the mu-calculus. For example, they have already been successfully used as a basis for an approach leading to a natural formula size game for the logic. While our main focus is introducing the new GTS, we also consider some applications to demonstrate its uses. For example, we consider a natural model transformation procedure that reduces model checking games to checking a single, fixed formula in the constructed models, and we also use the GTS to identify new alternative variants of the mu-calculus with PTime model checking.

A doctrinal approach to modal/temporal Heyting logic and non-determinism in processes

Abstract: The study of algebraic modelling of labelled non-deterministic concurrent processes leads us to consider a category LB , obtained from a complete meet-semilattice B and from B-valued equivalence relations. We prove that, if B has enough properties, then LB presents a two-fold internal logical structure, induced by two doctrines definable on it: one related to its families of subobjects and one to its families of regular subobjects. The first doctrine is Heyting and makes LB a Heyting category, the second one is Boolean. We will see that the difference between these two logical structures, namely the different behaviour of the negation operator, can be interpreted in terms of a distinction between non-deterministic and deterministic behaviours of agents able to perform computations in the context of the same process. Moreover, the sorted first-order logic naturally associated with LB can be extended to a modal/temporal logic, again using the doctrinal setting. Relations are also drawn to other computational models.

Kripke Completeness of Strictly Positive Modal Logics over Meet-semilattices with Operators.

Abstract: Our concern is the completeness problem for spi-logics, that is, sets of implications between strictly positive formulas built from propositional variables, conjunction and modal diamond operators. Originated in logic, algebra and computer science, spi-logics have two natural semantics: meet-semilattices with monotone operators providing Birkhoff-style calculi, and first-order relational structures (aka Kripke frames) often used as the intended structures in applications. Here we lay foundations for a completeness theory that aims to answer the question whether the two semantics define the same consequence relations for a given spi-logic.

Quantification in Some Non-normal Modal Logics

Abstract: This paper offers a semantic study in multi-relational semantics of quantified N-Monotonic modal logics with varying domains with and without the identity symbol. We identify conditions on frames to characterise Barcan and Ghilardi schemata and present some related completeness results. The characterisation of Barcan schemata in multi-relational frames with varying domains shows the independence of BF and CBF from well-known propositional modal schemata, an independence that does not hold with constant domains. This fact was firstly suggested for classical modal systems by Stolpe (Logic Journal of the IGPL 11(5), 557–575, 2003), but unfortunately that work used only models and not frames.

Graphical Sequent Calculi for Modal Logics

Abstract: The syntax of modal graphs is defined in terms of the continuous cut and broken cut following Charles Peirce's notation in the gamma part of his graphical logic of existential graphs. Graphical calculi for normal modal logics are developed based on a reformulation of the graphical calculus for classical propositional logic. These graphical calculi are of the nature of deep inference. The relationship between graphical calculi and sequent calculi for modal logics is shown by translations between graphs and modal formulas.

Intuitionistic Fuzzy Modal Logics

Abstract: The first step of the development of the idea of intuitionistic fuzziness (see [1]), was related to introducing an intuitionistic fuzzy interpretation of the classical (standard) modal operators “necessity” and “possibility” (see, e.g., [2, 3, 4, 5]). In the period 1988–1993, we defined eight new operators, extending the first two ones. In the end of last and in the beginning of this century, a lot of new operators were introduced. Here, we discuss the most interesting ones of them and study their basic properties.

Indicative Conditionals and Dynamic Epistemic Logic

Abstract: Recent ideas about epistemic modals and indicative conditionals in formal semantics have significant overlap with ideas in modal logic and dynamic epistemic logic. The purpose of this paper is to show how greater interaction between formal semantics and dynamic epistemic logic in this area can be of mutual benefit. In one direction, we show how concepts and tools from modal logic and dynamic epistemic logic can be used to give a simple, complete axiomatization of Yalcin's [16] semantic consequence relation for a language with epistemic modals and indicative conditionals. In the other direction, the formal semantics for indicative conditionals due to Kolodny and MacFarlane [9] gives rise to a new dynamic operator that is very natural from the point of view of dynamic epistemic logic, allowing succinct expression of dependence (as in dependence logic) or supervenience statements. We prove decidability for the logic with epistemic modals and Kolodny and MacFarlane's indicative conditional via a full and faithful computable translation from their logic to the modal logic K45.

A New Game Equivalence and its Modal Logic

Abstract: We revisit the crucial issue of natural game equivalences, and semantics of game logics based on these. We present reasons for investigating finer concepts of game equivalence than equality of standard powers, though staying short of modal bisimulation. Concretely, we propose a more finegrained notion of equality of "basic powers" which record what players can force plus what they leave to others to do, a crucial feature of interaction. This notion is closer to game-theoretic strategic form, as we explain in detail, while remaining amenable to logical analysis. We determine the properties of basic powers via a new representation theorem, find a matching "instantial neighborhood game logic", and show how our analysis can be extended to a new game algebra and dynamic game logic.

Intuitionistic modal logic based on neighborhood semantics without superset axiom

Abstract: In this paper we investigate certain systems of propositional intuitionistic modal logic defined semantically in terms of neighborhood structures. We discuss various restrictions imposed on those frames but our constant approach is to discard superset axiom. Such assumption allows us to think about specific modalities $\Delta, abla$ and new functor $\rightsquigarrow$ depending on the notion of maximal neighborhood. We show how it is possible to treat our models as bi-relational ones. We prove soundness and completeness of proposed axiomatization by means of canonical model. Moreover, we show that finite model property holds. Then we describe properties of bounded morphism, behavioral equivalence, bisimulation and n-bisimulation. Finally, we discuss further researches (like some interesting classical cases and particular topological issues).

Normal Modal Logics Determined by Aligned Clusters

Abstract: We consider the family of logics from NExt(KTB) which are determined by linear frames with reflexive and symmetric relation of accessibility. The condition of linearity in such frames was first defined in the paper [9]. We prove that the cardinality of the logics under consideration is uncountably infinite.

Cut-elimination for the modal Grzegorczyk logic via non-well-founded proofs

Abstract: We present a sequent calculus for the modal Grzegorczyk logic Grz allowing non-well-founded proofs and obtain the cut-elimination theorem for it by constructing a continuous cut-elimination mapping acting on these proofs.

Modal logic S4 as a paraconsistent logic with a topological semantics

Abstract: In this paper the propositional logic LTop is introduced, as an extension of classical propositional logic by adding a paraconsistent negation. This logic has a very natural interpretation in terms of topological models. The logic LTop is nothing more than an alternative presentation of modal logic S4, but in the language of a paraconsistent logic. Moreover, LTop is a logic of formal inconsistency in which the consistency and inconsistency operators have a nice topological interpretation. This constitutes a new proof of S4 as being “the logic of topological spaces”, but now under the perspective of paraconsistency. 1 Topology, Modal Logic and Paraconsistency The studies on the relationship between modal logic, topology and paraconsistency, have a relatively long history. By extending the Stone representation theorem for Boolean algebras, McKinsey and Tarski (see [14]) proved in 1944 that it is possible to characterize modal logic S4 by means of a topological semantics. Within this semantics, the necessity operator � and the possibility operator ♦ are interpreted as the interior and the closure topological operators, respectively, and so this result states that S4 is, in a certain sense, “the logic of topological spaces”. Moreover, they prove that S4 is semanticaly characterized by the real line (with the usual topology) or, in general, by any dense-in-itself separable metrizable space. Several variants and generalizations of McKinsey and Tarski’s result have been proposed in the literature (see, for instance, [18] and [12]). Semantics for Paraconsistent logics (that is, logics having a negation which produces some non-trivial contradictory theories) have been defined in topological terms by several authors. For instance, Mortensen studies in [16] some topological properties by means of paraconsistent and paracomplete logics. Goodman already proposes in [10] an “anti-intuitionistic logic” which is paraconsistent and it is endowed with a topological semantics. Along the same lines, Priest analyzes a paraconsistent negation obtained by dualizing the intuitionistic negation, defining a topological semantics for such negation (see [17]). From a broader perspective, Başkent proposes in [1] an interesting study of topological models for paraconsistency and paracompleteness. 2 Marcelo E. Coniglio and Leonardo Prieto-Sanabria By its turn, the relationship between paraconsistency and modal logic is also very close. Already in 1948, Jaśkowski presented in [11] his “discussive logic”, which is considered the first formal system for a paraconsistent logic, and it was formalized in terms of modalities. Beziau observes in [2] (see also [3]) that the operator ¬α def = ∼�α defines in modal logic S5 a paraconsistent negation (here, ∼ denotes the classical negation). This relation between modalities and paraconsistent negation was already observed by Beziau in 1998 (despite the paper was published only in 2006, see [4]), from the perspective of Kripke semantics, when defining the logic Z. However, already in 1987, de Araújo et al. observed in [9] that a Kripke-style semantics can be given for Sette’s 3valued paraconsistent logic P1, based on Kripke frames for the modal logic T. In that semantics, the formula ¬α (for the paraconsistent negation ¬ of P1) is interpreted exactly as the modal formula ∼�α. Beziau’s approach was generalized by Marcos in [13], showing that there is a close correspondence between non-degenerate modal logics and the paraconsistent logics known as logics of formal inconsistency (see Section 6). This paper contributes to this discussion by introducing a propositional logic, called LTop, which extends classical propositional logic with a paraconsistent negation. This logic has a very natural interpretation in terms of topological models which associate to each formula a set (not necessarily open or close). The classical connectives are interpreted as usual, and the paraconsistent negation is interpreted as the topological closure of the complement, in a dual form to the usual interpretation of the intuitionistic negation (namely, the interior of the complement). This topological interpretation of a paraconsistent negation is very natural, and it was already proposed in [10], [16] and [1]. Modalities � and ♦ can be defined in the language of LTop, which are interpreted as the interior and closure operator, respectively. As expected, the logic LTop is nothing more than an alternative presentation of modal logic S4, but in a language (and through a Hilbert calculus) corresponding to an extension of classical logic by means of a paraconsistent negation. It is worth noting that the logic Z in [4] introduces an axiomatization of S5 as an extension of classical logic with a paraconsistent negation, and so the present result is a kind of generalization of such result, provided that S5 can be obtained from S4 by adding an additional axiom. Additionally, it is proved that LTop is a logic of formal inconsistency (see Section 6) in which the consistency and inconsistency operators have a nice topological interpretation. This constitutes a new proof of S4 as being “the logic of topological spaces”, but now under the perspective of paraconsistent logics. It is finally shown that intuitionistic propositional logic can be interpreted in LTop through a very natural conservative translation. 2 The propositional logic LTop In this section, the propositional logic LTop will be introduced by means of a Hilbert calculus, with a modal-like notion of derivations. DEFINITION 1. Let V = � pn : n ≥ 1 � be a denumerable set of propositional variables. Given the propositional signature Σ = {∼ ,¬ ,→} let L be the language generated by the set V over the signature Σ. S4 as a paraconsistent logic 3 As suggested by the notation, ∼ and ¬ are unary connectives representing two different negations, while → is a binary connective which represents an implication (in the logic LTop to be defined below). The connective ∼ will represent a classical negation, while ¬ will represent a paraconsistent negation. The implication will be also classical. The following usual abbreviations can be introduced in the language L: (conj) α ∧ β def = ∼(α → ∼β) (disj) α ∨ β def = ∼α → β DEFINITION 2 (Propositional Logic LTop). The logic LTop is given by the Hilbert calculus over the language L defined by following the axioms and inference rules:

Modal extensions of Łukasiewicz logic for modelling coalitional power

Abstract: Modal logics for reasoning about the power of coalitions capture the notion of effectivity functions associated with game forms. The main goal of coalition logics is to provide formal tools for modeling the dynamics of a game frame whose states may correspond to different game forms. The two classes of effectivity functions studied are the families of playable and truly playable effectivity functions, respectively. In this paper we generalize the concept of effectivity function beyond the yes/no truth scale. This enables us to describe the situations in which the coalitions assess their effectivity in degrees, based on functions over the outcomes taking values in a finite Łukasiewicz chain. Then we introduce two modal extensions of Łukasiewicz finite-valued logic together with many-valued neighborhood semantics in order to encode the properties of many-valued effectivity functions associated with game forms. As our main results we prove completeness theorems for the two newly introduced modal logics.

Method and system for checking natural language in proof models of modal logic

Abstract: A system for checking natural language in proof models of modal logic includes parsing natural language for parts of speech mapping (POSM) into logical symbols and expressions, then proving the logical expression in the logic model checker (LCM) by modal logic. The LCM is applied to computer program validation and requirement document verification.

A Real-Valued Modal Logic

Abstract: A many-valued modal logic is introduced that combines the usual Kripke frame semantics of the modal logic K with connectives interpreted locally at worlds by lattice and group operations over the real numbers. A labelled tableau system is provided and a coNEXPTIME upper bound obtained for checking validity in the logic. Focussing on the modal-multiplicative fragment, the labelled tableau system is then used to establish completeness for a sequent calculus that admits cut-elimination and an axiom system that extends the multiplicative fragment of Abelian logic.

Doxastic Reasoning with Multi-Source Justifications based on Second Order Propositional Modal Logic

Abstract: In multi-agent systems, an agent generally forms her belief based on information from multiple sources, such as messages from other agents or perception of the external environment. While modal epistemic logic has been a standard formalism for reasoning about agent's belief, it lacks the expressive power for tracking information sources. Justification logic (JL) provides the missing expressivity by using justification terms to keep track of the belief formation process. However, because JL does not make a clear distinction between potential and actual evidence, the interpretation of justification formulas in JL turns out to be ambiguous. In this paper, we present a justification-based multi-source reasoning formalism built upon second-order propositional modal logic. Our framework not only inherits the source-tracking advantage of JL but also allows the distinction between the actual observation and simply potential admissibility of evidence.

On the Decidability of Certain Semi-Lattice Based Modal Logics

Abstract: Sequent calculi are proof systems that are exceptionally suitable for proving the decidability of a logic. Several relevance logics were proved decidable using a technique attributable to Curry and Kripke. Further enhancements led to a proof of the decidability of implicational ticket entailment by Bimbo and Dunn in [12, 13]. This paper uses a different adaptation of the same core proof technique to prove a group of positive modal logics (with disjunction but no conjunction) decidable.